Group #31897417 — subgroups

Subgroups — #31897417

Order 18176.

498 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588478	C₂	1	1	yes	maximal, characteristic, complemented
2	#22586791	C₂	1	2	yes	maximal, complemented
3	#22586791	C₂	1	2	yes	maximal, complemented
4	C₂×#17985721	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#3147	C₂	1	1	yes	maximal, characteristic
6	#17986023	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#17985764	—	2	2	no	—
10	D₄⋊(C₇₁⋊C₈)	—	2	4	no	—
11	D₄⋊(C₇₁⋊C₈)	—	2	4	no	—
12	#17985721	—	2	2	no	—
13	C₂×#135×C₇₁	—	2	2	no	—
14	#532×C₇₁	C₂²	1	2	yes	—
15	#495×C₇₁	—	2	2	no	—
16	#494×C₇₁	C₄	1	1	yes	characteristic
17	#493×C₇₁	C₄	1	1	yes	characteristic
18	#492×C₇₁	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#11765849	—	2	2	no	—
21	C₂²×C₇₁⋊C₈	—	2	1	no	—
22	C₂²×C₇₁⋊C₈	—	2	1	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	C₄×C₇₁⋊C₈	—	2	1	no	—
26	C₄×C₇₁⋊C₈	—	2	1	no	—
27	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
28	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
29	C₂×#41×C₇₁	—	2	2	no	—
30	#135×C₇₁	—	4	8	no	—
31	C₂×Q₁₆×C₇₁	—	2	2	no	—
32	C₂×SD₁₆×C₇₁	—	2	4	no	—
33	C₂×D₈×C₇₁	—	2	2	no	—
34	C₂²×C₈×C₇₁	D₄	1	2	yes	—
35	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
36	#127×C₇₁	C₂×C₄	1	2	yes	—
37	#124×C₇₁	—	2	2	no	—
38	#121×C₇₁	—	2	2	no	—
39	#121×C₇₁	—	2	2	no	—
40	C₄×D₄×C₇₁	—	2	2	no	—
41	C₄×D₄×C₇₁	—	2	2	no	—
42	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	D₄	1	2	yes	—
46	#11765849	—	4	4	no	—
47	C₂×C₇₁⋊C₈	—	2	4	no	—
48	C₂×C₇₁⋊C₈	—	2	2	no	—
49	C₂×C₇₁⋊C₈	—	2	2	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₂×C₇₁⋊C₈	—	4	2	no	—
52	#41×C₇₁	—	4	8	no	—
53	C₂×Q₈×C₇₁	—	2	2	no	—
54	C₂×D₄×C₇₁	—	2	2	no	—
55	C₂×D₄×C₇₁	—	2	2	no	—
56	C₂×D₄×C₇₁	—	2	2	no	—
57	C₂×D₄×C₇₁	—	4	2	no	—
58	C₂²×C₄×C₇₁	—	4	2	no	—
59	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
62	Q₁₆×C₇₁	—	4	4	no	—
63	SD₁₆×C₇₁	—	4	8	no	—
64	D₈×C₇₁	—	4	4	no	—
65	C₂×C₈×C₇₁	—	2	4	no	—
66	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
68	C₄⋊C₄×C₇₁	—	2	2	no	—
69	C₄⋊C₄×C₇₁	—	2	2	no	—
70	C₂²⋊C₄×C₇₁	—	4	2	no	—
71	C₂²⋊C₄×C₇₁	—	4	2	no	—
72	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
76	C₂³×C₇₁	—	4	2	no	—
77	C₂³×C₇₁	#124	1	1	yes	characteristic
78	Q₈×C₇₁	—	4	4	no	—
79	D₄×C₇₁	—	4	4	no	—
80	D₄×C₇₁	—	4	4	no	—
81	D₄×C₇₁	—	4	4	no	—
82	D₄×C₇₁	—	8	4	no	—
83	C₂×C₄×C₇₁	—	2	2	no	—
84	C₂×C₄×C₇₁	—	2	2	no	—
85	C₂×C₄×C₇₁	—	4	2	no	—
86	C₂×C₄×C₇₁	—	8	4	no	—
87	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
93	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	4	yes	—
94	C₈×C₇₁	—	2	2	no	—
95	C₈×C₇₁	—	2	2	no	—
96	C₇₁⋊C₈	—	2	4	no	—
97	C₇₁⋊C₈	—	4	2	no	—
98	C₇₁⋊C₈	—	8	2	no	—
99	C₂²×C₇₁	—	2	1	no	—
100	C₂²×C₇₁	—	2	1	no	—
101	C₂²×C₇₁	—	8	2	no	—
102	C₂²×C₇₁	—	8	2	no	—
103	C₂²×C₇₁	—	8	2	no	—
104	C₂²×C₇₁	#390	1	1	yes	characteristic
105	C₂²×C₇₁	#490	1	2	yes	—
106	C₄×C₇₁	—	2	1	no	—
107	C₄×C₇₁	—	2	1	no	—
108	C₄×C₇₁	—	8	2	no	—
109	C₄×C₇₁	#418	1	2	yes	—
110	C₄×C₇₁	#443	1	2	yes	—
111	C₄×C₇₁	#444	1	2	yes	—
112	C₄×C₇₁	#496	1	2	yes	—
113	#9860	—	71	1	no	maximal, Sylow
114	C₂×C₇₁	—	2	2	no	—
115	C₂×C₇₁	—	16	2	no	—
116	C₂×C₇₁	#1961	1	1	yes	characteristic
117	C₂×C₇₁	#1974	1	1	yes	characteristic
118	C₂×C₇₁	#1975	1	1	yes	characteristic
119	#3147	D₇₁	1	1	yes	characteristic
120	#1837	—	71	1	no	—
121	#1492	—	71	2	no	—
122	#1492	—	71	2	no	—
123	C₂×#322	—	71	1	no	—
124	C₇₁	#9860	1	1	yes	characteristic, complemented, Sylow
125	C₂×#135	—	2	2	no	—
126	#532	C₂×D₇₁	1	2	yes	—
127	#495	—	2	2	no	—
128	#494	Dic₇₁	1	1	yes	characteristic
129	#493	Dic₇₁	1	1	yes	characteristic
130	#492	Dic₇₁	1	2	yes	—
131	#423	—	71	2	no	—
132	C₂×C₄×C₈	—	71	1	no	—
133	C₂×C₄×C₈	—	71	1	no	—
134	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
135	#341	—	142	2	no	—
136	D₄⋊C₈	—	142	4	no	—
137	D₄⋊C₈	—	142	4	no	—
138	#322	—	142	2	no	—
139	C₂×#41	—	2	2	no	—
140	#135	—	4	8	no	—
141	C₂×Q₁₆	—	2	2	no	—
142	C₂×SD₁₆	—	2	4	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×M₁₆	—	142	2	no	—
145	C₂²×C₈	—	142	1	no	—
146	C₂²×C₈	—	142	1	no	—
147	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
148	C₄⋊Q₈	C₂×Dic₇₁	1	2	yes	—
149	#127	C₂×Dic₇₁	1	2	yes	—
150	#124	—	2	2	no	—
151	#121	—	2	2	no	—
152	#121	—	2	2	no	—
153	C₄×D₄	—	2	2	no	—
154	C₄×D₄	—	2	2	no	—
155	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
156	C₄⋊C₈	—	142	2	no	—
157	C₄⋊C₈	—	142	2	no	—
158	C₄×C₈	—	71	2	no	—
159	C₄×C₈	—	71	2	no	—
160	C₄×C₈	—	142	1	no	—
161	C₄×C₈	—	142	1	no	—
162	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
163	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
165	#41	—	4	8	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
173	C₂²×C₄	#11765864	1	1	yes	characteristic
174	C₂²×C₄	#11765864	1	1	yes	characteristic
175	Q₁₆	—	4	4	no	—
176	SD₁₆	—	4	8	no	—
177	D₈	—	4	4	no	—
178	M₁₆	—	284	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	284	2	no	—
181	C₂×C₈	—	284	2	no	—
182	C₂×C₈	—	142	2	no	—
183	C₂×C₈	—	142	4	no	—
184	C₂×C₈	—	142	2	no	—
185	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
186	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
192	C₄²	D₄×D₇₁	1	1	yes	characteristic
193	C₄²	D₄×D₇₁	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
195	C₂³	—	4	2	no	—
196	C₂³	#14556115	1	1	yes	characteristic
197	Q₈	—	4	4	no	—
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	4	4	no	—
201	D₄	—	8	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	8	4	no	—
206	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
207	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
208	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
209	C₂×C₄	#14556099	1	1	yes	characteristic
210	C₂×C₄	#14556099	1	1	yes	characteristic
211	C₂×C₄	#14556124	1	1	yes	characteristic
212	C₂×C₄	#14556154	1	4	yes	—
213	C₈	—	2	2	no	—
214	C₈	—	2	2	no	—
215	C₈	—	284	2	no	—
216	C₈	—	568	2	no	—
217	C₈	—	142	4	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	8	2	no	—
221	C₂²	—	8	2	no	—
222	C₂²	—	8	2	no	—
223	C₂²	#17985902	1	1	yes	characteristic
224	C₂²	#17986363	1	2	yes	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	8	2	no	—
228	C₄	#17986004	1	2	yes	—
229	C₄	#17986116	1	2	yes	—
230	C₄	#17986121	1	2	yes	—
231	C₄	#17986386	1	2	yes	—
232	C₂	—	2	2	no	—
233	C₂	—	16	2	no	—
234	C₂	#22589165	1	1	yes	characteristic
235	C₂	#22589229	1	1	yes	characteristic
236	C₂	#22589234	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588478

C₂

yes

maximal, characteristic, complemented

#22586791

C₂

yes

maximal, complemented

#22586791

C₂

yes

maximal, complemented

C₂×#17985721

C₂

yes

maximal, characteristic, complemented

C₇₁×#3147

C₂

yes

maximal, characteristic

#17986023

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

#17985764

—

D₄⋊(C₇₁⋊C₈)

—

D₄⋊(C₇₁⋊C₈)

—

#17985721

—

C₂×#135×C₇₁

—

#532×C₇₁

C₂²

yes

—

#495×C₇₁

—

#494×C₇₁

C₄

yes

characteristic

#493×C₇₁

C₄

yes

characteristic

#492×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂×#41×C₇₁

—

#135×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×SD₁₆×C₇₁

—

C₂×D₈×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄⋊Q₈×C₇₁

C₂×C₄

yes

—

#127×C₇₁

C₂×C₄

yes

—

#124×C₇₁

—

#121×C₇₁

—

#121×C₇₁

—

C₄×D₄×C₇₁

—

C₄×D₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

#41×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

C₂²×C₄×C₇₁

yes

C₂²×C₄×C₇₁

yes

Q₁₆×C₇₁

—

SD₁₆×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

—

C₂³×C₇₁

#124

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

100

C₂²×C₇₁

—

101

C₂²×C₇₁

—

102

C₂²×C₇₁

—

103

C₂²×C₇₁

—

104

C₂²×C₇₁

#390

yes

characteristic

105

C₂²×C₇₁

#490

yes

—

106

C₄×C₇₁

—

107

C₄×C₇₁

—

108

C₄×C₇₁

—

109

C₄×C₇₁

#418

yes

—

110

C₄×C₇₁

#443

yes

—

111

C₄×C₇₁

#444

yes

—

112

C₄×C₇₁

#496

yes

—

113

#9860

—

maximal, Sylow

114

C₂×C₇₁

—

115

C₂×C₇₁

—

116

C₂×C₇₁

#1961

yes

characteristic

117

C₂×C₇₁

#1974

yes

characteristic

118

C₂×C₇₁

#1975

yes

characteristic

119

#3147

D₇₁

yes

characteristic

120

#1837

—

121

#1492

—

122

#1492

—

123

C₂×#322

—

124

C₇₁

#9860

yes

characteristic, complemented, Sylow

125

C₂×#135

—

126

#532

C₂×D₇₁

yes

—

127

#495

—

128

#494

Dic₇₁

yes

characteristic

129

#493

Dic₇₁

yes

characteristic

130

#492

Dic₇₁

yes

—

131

#423

—

132

C₂×C₄×C₈

—

133

C₂×C₄×C₈

—

134

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

135

#341

—

142

—

136

D₄⋊C₈

—

142

—

137

D₄⋊C₈

—

142

—

138

#322

—

142

—

139

C₂×#41

—

140

#135

—

141

C₂×Q₁₆

—

142

C₂×SD₁₆

—

143

C₂×D₈

—

144

C₂×M₁₆

—

142

—

145

C₂²×C₈

—

142

—

146

C₂²×C₈

—

142

—

147

C₂²×C₈

C₂²⋊D₇₁

yes

—

148

C₄⋊Q₈

C₂×Dic₇₁

yes

—

149

#127

C₂×Dic₇₁

yes

—

150

#124

—

151

#121

—

152

#121

—

153

C₄×D₄

—

154

C₄×D₄

—

155

C₂×C₄²

C₂²×D₇₁

yes

characteristic

156

C₄⋊C₈

—

142

—

157

C₄⋊C₈

—

142

—

158

C₄×C₈

—

159

C₄×C₈

—

160

C₄×C₈

—

142

—

161

C₄×C₈

—

142

—

162

C₄×C₈

C₂×Dic₇₁

yes

characteristic

163

C₄×C₈

C₂×Dic₇₁

yes

characteristic

164

C₄×C₈

C₂²⋊D₇₁

yes

—

165

#41

—

166

C₂×Q₈

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂×D₄

—

170

C₂×D₄

—

171

C₂²×C₄

—

172

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

173

C₂²×C₄

#11765864

yes

characteristic

174

C₂²×C₄

#11765864

yes

characteristic

175

Q₁₆

—

176

SD₁₆

—

177

D₈

—

178

M₁₆

—

284

—

179

C₂×C₈

—

180

C₂×C₈

—

284

—

181

C₂×C₈

—

284

—

182

C₂×C₈

—

142

—

183

C₂×C₈

—

142

—

184

C₂×C₈

—

142

—

185

C₂×C₈

C₂²⋊Dic₇₁

yes

—

186

C₂×C₈

C₂²⋊Dic₇₁

yes

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₂²⋊C₄

—

190

C₂²⋊C₄

—

191

C₄²

C₂²×Dic₇₁

yes

characteristic

192

C₄²

D₄×D₇₁

yes

characteristic

193

C₄²

D₄×D₇₁

yes

characteristic

194

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

195

C₂³

—

196

C₂³

#14556115

yes

characteristic

197

Q₈

—

198

D₄

—

199

D₄

—

200

D₄

—

201

D₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

207

C₂×C₄

D₄×Dic₇₁

yes

characteristic

208

C₂×C₄

D₄×Dic₇₁

yes

characteristic

209

C₂×C₄

#14556099

yes

characteristic

210

C₂×C₄

#14556099

yes

characteristic

211

C₂×C₄

#14556124

yes

characteristic

212

C₂×C₄

#14556154

yes

—

213

C₈

—

214

C₈

—

215

C₈

—

284

—

216

C₈

—

568

—

217

C₈

—

142

—

218

C₂²

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

#17985902

yes

characteristic

224

C₂²

#17986363

yes

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

#17986004

yes

—

229

C₄

#17986116

yes

—

230

C₄

#17986121

yes

—

231

C₄

#17986386

yes

—

232

C₂

—

233

C₂

—

234

C₂

#22589165

yes

characteristic

235

C₂

#22589229

yes

characteristic

236

C₂

#22589234

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting, prime radical	C₇₁×#3147	C₂
socle	C₂²×C₇₁	#390
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#3147	D₇₁
hypocenter	C₇₁	#9860
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

subgroup

G/H

center

C₂×C₄

#14556124

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting, prime radical

C₇₁×#3147

C₂

socle

C₂²×C₇₁

#390

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

N	H ≅ G/N	type
1. #22588478	C₂	semidirect
2. #22586791	C₂	semidirect
3. #22586791	C₂	semidirect
4. C₂×#17985721	C₂	semidirect
124. C₇₁	#9860	semidirect

H ≅ G/N

type

1. #22588478

C₂

semidirect

2. #22586791

C₂

semidirect

3. #22586791

C₂

semidirect

4. C₂×#17985721

C₂

semidirect

124. C₇₁

#9860

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22588478	2
#22586791	2
C₂×#17985721	2
C₇₁×#3147	2
#9860	71

subgroup

index

C₇₁×#3147

Subgroups — #31897417

Conjugacy classes of subgroups

Complements

Maximal subgroups